Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from William James, via the Furman University Mathematical Quotations Server. Background photo courtesy of Joe Maggie-Me (CC BY 2.0) via Flickr.
Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Erlina Ronda, The fun in learning mathematics is in the challenge. Background photo courtesy of USAG-Humphreys (CC BY 2.0) via flickr.
Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Howard Eves, An Introduction to the History of Mathematics. Background photo courtesy of Brenda Clarke (CC BY 2.0) via flickr.
During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.
“Can you tell me how to get to twelve?”
My five year old begins, “You could take two fives and add a two.”
“Take sixty and divide it into five parts,” my nearly-seven year old says.
“You could do two tens and then take away a five and a three,” my younger son adds.
Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.
I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.
Instead, I played baseball.
The best way for children to build mathematical fluency is through conversation. For more ideas on discussion-based math, check out these posts:
Learning the Math Facts
For more help with learning and practicing the basic arithmetic facts, try these tips and math games:
- Memorizing the Math Facts
- Math Facts Are like Learning to Type
- Game: Times Tac Toe
- Contig Game: Master Your Math Facts
feature photo above by Junya Ogura via flickr (CC BY 2.0)
A couple of weeks ago, James Tanton launched a wonderful resource: a free online course devoted to quadratic equations. (And he promises more topics to come.)
Kitten and I have been working through the lessons, and she loves it!
We’re skimming through pre-algebra in our regular lessons, but she has enjoyed playing around with simple algebra since she was in kindergarten. She has a strong track record of thinking her way through math problems, and earlier this year she invented her own method for solving systems of equations with two unknowns. I would guess her background is approximately equal to an above-average algebra 1 student near the end of the first semester.
After few lessons of Tanton’s course, she proved — within the limits of experimental error — that a catenary (the curve formed by a hanging chain) cannot be described by a quadratic equation. Last Friday, she easily solved the following equations:
and (though it took a bit more thought):
We’ve spent less than half an hour a day on the course, as a supplement to our AoPS Pre-Algebra textbook. We watch each video together, pausing occasionally so she can try her hand at an equation before listening to Tanton’s explanation. Then (usually the next day) she reads the lesson and does the exercises on her own. So far, she hasn’t needed the answers in the Companion Guide to Quadratics, but she did use the “Dots on a Circle” activity — and knowing that she has the answers available helps her feel more independent.
Do you enjoy math? I hope so! If not, browsing this post just may change your mind. Welcome to the Math Teachers At Play blog carnival — a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college.
Let the mathematical fun begin!
By tradition, we start the carnival with a puzzle in honor of our 62nd edition:
An Archimedean solid is a polyhedron made of two or more types of regular polygons meeting in identical vertices. A rhombicosidodecahedron (see image above) has 62 sides: triangles, squares, and pentagons.
- How many of each shape does it take to make a rhombicosidodecahedron?
My math club students had fun with a Polyhedra Construction Kit. Here’s how to make your own:
- Collect a bunch of empty cereal boxes. Cut the boxes open to make big sheets of cardboard.
- Print out the template page (→) and laminate. Cut out each polygon shape, being sure to include the tabs on the sides.
- Turn your cardboard brown-side-up and trace around the templates, making several copies of each polygon. I recommend 20 each of the pentagon and hexagon, 40 each of the triangle and square.
- Draw the dark outline of each polygon with a ballpoint pen, pressing hard to score the cardboard so the tabs will bend easily.
- Cut out the shapes, being careful around the tabs.
- Use small rubber bands to connect the tabs. Each rubber band will hold two tabs together, forming one edge of a polyhedron.
So, for instance, it takes six squares and twelve rubber bands to make a cube. How many different polyhedra (plural of polyhedron) will you make?
- Can you build a rhombicosidodecahedron?
And now, on to the main attraction: the 62 blog posts. Many of the following articles were submitted by their authors; others were drawn from the immense backlog in my blog reader. If you’d like to skip directly to your area of interest, here’s a quick Table of Contents:
- Early Learning Activities
- Elementary Exploration and Middle School Mastery
- Adventures in Basic Algebra and Geometry
- Advanced Mathematical Endeavors
- Puzzling Recreations
- Teaching Tips
As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary and dangerous than sailing the high seas!
But most parents and educators don’t present math this way. They just want the children to develop their mathematical skills rather than going for something more nebulous, like the mathematical state of mind.
Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive.
Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.
— Maria Droujkova and Yelena McManaman
Adventurous Math For the Playground Set (Scientific American online)
[Photo by AlphaTangoBravo / Adam Baker via flickr.]
One reason to study algebra: because it’s a building block. And just as it was really hard at first to get those blocks to do what you wanted them to do, so also it can be really hard at first to get algebra to work. But if you persevere, who knows what you might build someday?
Algebra is the beginning of a journey that gives you the skills to solve more complex problems.
So, try not to think of Algebra as a boring list of rules and procedures to memorize. Consider algebra as a gateway to exploring the world around us all.
— Jason Gibson
Why Study Algebra?
Food for thought:
Imagine that you wanted your children to learn the names of all their cousins, aunts and uncles. But you never actually let them meet or play with them. You just showed them pictures of them, and told them to memorize their names.
Each day you’d have them recite the names, over and over again. You’d say, “OK, this is a picture of your great-aunt Beatrice. Her husband was your great-uncle Earnie. They had three children, your uncles Harpo, Zeppo, and Gummo. Harpo married your aunt Leonie … yadda, yadda, yadda.
— Brian Foley
Times Tables – The Worst Way to Teach Multiplication
On the other hand, if you want your children to develop relationships with the numbers, to learn the math facts naturally, then be sure to tell lots of math stories. And when you are ready to focus on multiplication, be sure to study the patterns and relationships within the times tables.
I finally got around to reading a bit of the backlog in my Google Reader. I love Malke’s blog!
I used to think that math was some kind of inaccessible, abstract magic trick, a sort of in-joke that excluded us common folk, but now I realize that math is completely not that at all. The reality of math as most of us know it is like that story where three men are standing in a dark room touching different parts of an elephant. None of them has the full picture because they’re only perceiving individual elements of the whole animal.
The reality, I’m discovering, is that math is just like that elephant: a large, expansive, three-dimensional, intelligent, sensitive, expressive creature.
The problem is that most of us have been standing around in that dark room since about kindergarten, grasping its tail, thinking “this is what math is and, personally, I don’t think it’s for me.” We’ve been blind to the larger, incredibly beautiful picture that would emerge if only we would turn on the lights and open our eyes.
Check out Dan’s interesting semi-philosophical discussion of the meaning and importance of abstraction:
- [LOA] The Ladder of Abstraction, Part One Of Probably A Lot
- [LOA] The Place Where Language And Math Make Friends
- [LOA] Abstracting Abstraction
- to be continued…
The physical five oranges goes up the ladder to the picture of the five oranges which goes up to the representation of the five oranges as a numeral.
This points in the direction of a definition of abstraction: when we abstract we voluntarily ignore details of a context, so that we can accomplish a goal.
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
I had the most beautiful set of theories you ever knew when I started out as a schoolma’am, but every one of them has failed me at some pinch or another.
“We have 4 equations and only 4 unknowns so that gives us a fighting chance of actually solving it.”
— My daughter’s Calculus III teacher
“Of course, he was doing an easy problem compared to the homework. ”
— My daughter, Niner
As I continue to polish the manuscript for my math games book, I’ve been looking for short quotations to put at the beginning of each chapter. I’ve gathered a lot of math quotations over the years, from my own reading and from quote-collection websites. But there’s a problem with using most of these in a book, since to do it right I would have to dig up the original source of each quote and then write a letter to the publisher for permission to use it. And pay a fee that, depending on the publisher’s sense of self-importance, can run into the hundreds of dollars. Bother!
So I went digging around my rss reader to see what sort of inspiration I could find. Bloggers love to be quoted, right? And most of them are happy to give permission via email, which makes my job ever so much easier.
Here are some of the gems I’m considering. I’d love to hear your favorite quotes from math bloggers, too — or favorite passages from your own blog. Please comment!
It’s amazing that this vision of math as “getting to the right answer on your first try” even exists. I have to make, unmake, remake so many mistakes to get where I’m going. I think all mathematicians work that way.
Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!
Seen at kitchen table math, the sequel:
I can recall the deep satisfaction I felt on the all-too-rare occasions at school when the concepts or formulas fell into place. It seemed an entirely different discipline from writing, where something arises from a blank page through a combination of hard work and patience, with a sliver of creativity.
With math, the experience is more like discovering something that’s always existed and finally decided to stop playing hard-to-get.
We have an interesting discussion going in the comments on The Problem with Manipulatives. I mentioned a vague memory of a quotation. Now I’ve found the source.
Originally published in 1970:
The continuing hullabaloo about the “new math” has given many a parent a false impression. What was formerly a dull way of teaching mathematics by rote, so goes the myth, has suddenly been replaced by a marvelous new technique that is achieving miraculous results throughout the nation’s public schools.
I wish it were true — even if only to the extent implied by entertainer (and math teacher) Tom Lehrer in his delightfully whimsical recording on “The New Math”:
“In the new approach, as you know, the important thing is to understand what you’re doing, rather than to get the right answer.”
… Indeed, there is something to be said for the old math when taught by a poorly trained teacher. He can, at least, get across the fundamental rules of calculation without too much confusion. The same teacher trying to teach new math is apt to get across nothing at all…
Unfortunately, I can’t embed the Tom Lehrer song Gardner mentioned, due to copyright restrictions, but here’s a link to YouTube:
I don’t love math nearly as much as I pretend I do when I’m teaching it or blogging about it or trying to enthuse my kids.
I just believe — ever since an eye-opening university-level Mathematics in Perspective course — that math is taught VERY badly, bumbled and fumbled and as a result we have this societal fear of what is, essentially, a great big GAME.
See related post — Quotations XXV: Math is a Game
When a kid is feeling bad about being stuck with a problem, or just very anxious, I sometimes ask him to make as many mistakes as he can, and as outrageous as he can. Laughter happens (which is valuable by itself, and not only for the mood — deep breathing brings oxygen to the brain). Then the kid starts making mistakes. In the process, features of the problem become much clearer, and in many cases a way to a solution presents itself.
Does It Work?
Recognize that every math program, whether more traditionally skill-based or reform-oriented (more problem-solving, projects, less drill) has its merits and its weaknesses. Whether you believe there is too much emphasis on basic facts (less likely!), or not enough, you can supplement with the myriad of resources on the web.
Click through to the original post for a counting puzzle, plenty of advice on helping with your child’s math homework, useful math links, and a couple of “cute 3-year-old” stories.
And remember that one of the best ways to supplement any math program is by playing games.
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
It’s like asking why Beethoven’s Ninth symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
Important note: times tables are not math. Math doesn’t need to be made fun; it already is fun. Memorizing your times tables is a rote activity, it requires a fair bit of repetition for most, and it may need to be made fun. Just saying.
It’s a great game! Do click over to Dan’s blog and check it out:
And while we’re on the topic of times tables, Maria posted an article, too:
There’s a tendency for adults to label the math that they can do (such as identifying patterns, choosing between competing offers in a supermarket, and challenging statistics published by the government) as “common sense” and labeling everything they can’t do as “math” — so that being bad at math becomes a self-fulfilling prophecy.
Still a couple of slots left in Sol’s book giveaway. I’ll be hosting a giveaway soon, too — so watch this spot!
July 27th is Alex’s birthday. She shares it with Johann Bernoulli, an irascible mathematician from the late 17th century. This coincidence intrigued her enough that she wrote a research paper on Johann and his mathematical brother, titled “Jeering Jacob and Jealous Johann.”
Of course, to make the alliteration work, she had to mispronounce Johann’s name — but she figured he kinda deserved that. Read the historical tidbits below to find out why one writer said the Bernoulli brothers were “the kind of people who give arrogance a bad name.”*
[Photo by Micah Sittig.]
I used to fill the margins of my math newsletter with quotations and tidbits of math history. Here are some quotes from the July/August 1999 issue on probability, along with a few others I’ve stumbled on while browsing the internet.
No knowledge of probabilities helps us to know what conclusions are true. There is no direct relation between the truth of a proposition and its probability.
The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there’s a 90% probability you’ll get it wrong.
The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.
Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.
The entire discussion (article and comments) is well worth reading:
You may also enjoy:
- Math Facts Are like Learning to Type
- Math Facts: 5 Minutes a Day
- Cute Math Facts for Visual Thinkers
- Contig Game: Master Your Math Facts